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Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps).

Research & information: general --- Mathematics & science --- Lévy processes --- non-random overshoots --- skip-free random walks --- fluctuation theory --- scale functions --- capital surplus process --- dividend payment --- optimal control --- capital injection constraint --- spectrally negative Lévy processes --- reflected Lévy processes --- first passage --- drawdown process --- spectrally negative process --- dividends --- de Finetti valuation objective --- variational problem --- stochastic control --- optimal dividends --- Parisian ruin --- log-convexity --- barrier strategies --- adjustment coefficient --- logarithmic asymptotics --- quadratic programming problem --- ruin probability --- two-dimensional Brownian motion --- spectrally negative Lévy process --- general tax structure --- first crossing time --- joint Laplace transform --- potential measure --- Laplace transform --- first hitting time --- diffusion-type process --- running maximum and minimum processes --- boundary-value problem --- normal reflection --- Sparre Andersen model --- heavy tails --- completely monotone distributions --- error bounds --- hyperexponential distribution --- reflected Brownian motion --- linear diffusions --- drawdown --- Segerdahl process --- affine coefficients --- spectrally negative Markov process --- hypergeometric functions --- capital injections --- bankruptcy --- reflection and absorption --- Pollaczek–Khinchine formula --- scale function --- Padé approximations --- Laguerre series --- Tricomi–Weeks Laplace inversion

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The modeling and processing of empirical data is one of the main subjects and goals of statistics. Nowadays, with the development of computer science, the extraction of useful and often hidden information and patterns from data sets of different volumes and complex data sets in warehouses has been added to these goals. New and powerful statistical techniques with machine learning (ML) and data mining paradigms have been developed. To one degree or another, all of these techniques and algorithms originate from a rigorous mathematical basis, including probability theory and mathematical statistics, operational research, mathematical analysis, numerical methods, etc. Popular ML methods, such as artificial neural networks (ANN), support vector machines (SVM), decision trees, random forest (RF), among others, have generated models that can be considered as straightforward applications of optimization theory and statistical estimation. The wide arsenal of classical statistical approaches combined with powerful ML techniques allows many challenging and practical problems to be solved. This Special Issue belongs to the section “Mathematics and Computer Science”. Its aim is to establish a brief collection of carefully selected papers presenting new and original methods, data analyses, case studies, comparative studies, and other research on the topic of statistical data modeling and ML as well as their applications. Particular attention is given, but is not limited, to theories and applications in diverse areas such as computer science, medicine, engineering, banking, education, sociology, economics, among others. The resulting palette of methods, algorithms, and applications for statistical modeling and ML presented in this Special Issue is expected to contribute to the further development of research in this area. We also believe that the new knowledge acquired here as well as the applied results are attractive and useful for young scientists, doctoral students, and researchers from various scientific specialties.

mathematical competency --- assessment --- machine learning --- classification and regression tree --- CART ensembles and bagging --- ensemble model --- multivariate adaptive regression splines --- cross-validation --- dam inflow prediction --- long short-term memory --- wavelet transform --- input predictor selection --- hyper-parameter optimization --- brain-computer interface --- EEG motor imagery --- CNN-LSTM architectures --- real-time motion imagery recognition --- artificial neural networks --- banking --- hedonic prices --- housing --- quantile regression --- data quality --- citizen science --- consensus models --- clustering --- Gower’s interpolation formula --- Gower’s metric --- mixed data --- multidimensional scaling --- classification --- data-adaptive kernel functions --- image data --- multi-category classifier --- predictive models --- support vector machine --- stochastic gradient descent --- damped Newton --- convexity --- METABRIC dataset --- breast cancer subtyping --- deep forest --- multi-omics data --- categorical data --- similarity --- feature selection --- kernel density estimation --- non-linear optimization --- kernel clustering --- n/a --- Gower's interpolation formula --- Gower's metric

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This volume is a collection of investigations involving the theory and applications of the various tools and techniques of mathematical analysis and analytic number theory, which are remarkably widespread in many diverse areas of the mathematical, biological, physical, chemical, engineering, and statistical sciences. It contains invited and welcome original as well as review-cum-expository research articles dealing with recent and new developments on the topics of mathematical analysis and analytic number theory as well as their multidisciplinary applications.

Research & information: general --- Mathematics & science --- subordination --- functions with positive real part --- reciprocals --- univalent functions --- starlikeness --- convexity --- close-to-convexity --- hyper-Bessel functions --- Hardy space --- distribution --- fractional Laplacian --- Riesz fractional derivative --- delta sequence --- convolution --- subordinations --- starlike functions --- convex functions --- close-to-convex functions --- cardioid domain --- Hankel determinant --- m-fold symmetric functions --- harmonic univalent functions --- with symmetric conjecture point --- integral expressions --- coefficient estimates --- distortion --- umbral methods --- harmonic numbers --- special functions --- integral representations --- laplace and other integral transforms --- analytic functions --- quasi-Hadamard --- differential operator --- closure property --- riemann zeta function --- asymptotics --- exponential sums --- multivalent functions --- q-Ruschweyh differential operator --- q-starlike functions --- circular domain --- q-Bernardi integral operator --- Bessel functions --- Appell–Bessel functions --- generating functions --- Chebyshev polynomials --- Euler sums --- Catalan’s constant --- Trigamma function --- integral representation --- closed form --- ArcTan and ArcTanh functions --- partial fractions --- Lambert series --- cotangent sum --- modular transformation --- Dedekind sum --- lemniscate of Bernoulli Hankel determinant --- determinant --- inverse --- Mersenne number --- periodic tridiagonal Toeplitz matrix --- Sherman-Morrison-Woodbury formula --- Fibonacci number --- Lucas number --- Toeplitz matrix --- Hankel matrix --- univalent function --- second Hankel determinant --- bi-close-to-convex functions --- gamma function and its extension --- Pochhammer symbol and its extensions --- hypergeometric function and its extensions --- τ-Gauss hypergeometric function and its extensions --- τ-Kummer hypergeometric function --- Fox-Wright function --- p-valent analytic function --- Hadamard product --- q-integral operator --- generalized Lupaş operators --- q analogue --- Korovkin’s type theorem --- convergence theorems --- Voronovskaya type theorem --- starlike function --- subordinate --- Janowski functions --- conic domain --- q-convex functions --- q-close-to-convex functions --- theta-function identities --- multivariable R-functions --- Jacobi’s triple-product identity --- Ramanujan’s theta functions --- q-product identities --- Euler’s pentagonal number theorem --- Rogers-Ramanujan continued fraction --- Rogers-Ramanujan identities --- combinatorial partition-theoretic identities --- Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities --- Schur’s second partition theorem --- subordination --- functions with positive real part --- reciprocals --- univalent functions --- starlikeness --- convexity --- close-to-convexity --- hyper-Bessel functions --- Hardy space --- distribution --- fractional Laplacian --- Riesz fractional derivative --- delta sequence --- convolution --- subordinations --- starlike functions --- convex functions --- close-to-convex functions --- cardioid domain --- Hankel determinant --- m-fold symmetric functions --- harmonic univalent functions --- with symmetric conjecture point --- integral expressions --- coefficient estimates --- distortion --- umbral methods --- harmonic numbers --- special functions --- integral representations --- laplace and other integral transforms --- analytic functions --- quasi-Hadamard --- differential operator --- closure property --- riemann zeta function --- asymptotics --- exponential sums --- multivalent functions --- q-Ruschweyh differential operator --- q-starlike functions --- circular domain --- q-Bernardi integral operator --- Bessel functions --- Appell–Bessel functions --- generating functions --- Chebyshev polynomials --- Euler sums --- Catalan’s constant --- Trigamma function --- integral representation --- closed form --- ArcTan and ArcTanh functions --- partial fractions --- Lambert series --- cotangent sum --- modular transformation --- Dedekind sum --- lemniscate of Bernoulli Hankel determinant --- determinant --- inverse --- Mersenne number --- periodic tridiagonal Toeplitz matrix --- Sherman-Morrison-Woodbury formula --- Fibonacci number --- Lucas number --- Toeplitz matrix --- Hankel matrix --- univalent function --- second Hankel determinant --- bi-close-to-convex functions --- gamma function and its extension --- Pochhammer symbol and its extensions --- hypergeometric function and its extensions --- τ-Gauss hypergeometric function and its extensions --- τ-Kummer hypergeometric function --- Fox-Wright function --- p-valent analytic function --- Hadamard product --- q-integral operator --- generalized Lupaş operators --- q analogue --- Korovkin’s type theorem --- convergence theorems --- Voronovskaya type theorem --- starlike function --- subordinate --- Janowski functions --- conic domain --- q-convex functions --- q-close-to-convex functions --- theta-function identities --- multivariable R-functions --- Jacobi’s triple-product identity --- Ramanujan’s theta functions --- q-product identities --- Euler’s pentagonal number theorem --- Rogers-Ramanujan continued fraction --- Rogers-Ramanujan identities --- combinatorial partition-theoretic identities --- Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities --- Schur’s second partition theorem

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• Reference Manager
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Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps).

Research & information: general --- Mathematics & science --- Lévy processes --- non-random overshoots --- skip-free random walks --- fluctuation theory --- scale functions --- capital surplus process --- dividend payment --- optimal control --- capital injection constraint --- spectrally negative Lévy processes --- reflected Lévy processes --- first passage --- drawdown process --- spectrally negative process --- dividends --- de Finetti valuation objective --- variational problem --- stochastic control --- optimal dividends --- Parisian ruin --- log-convexity --- barrier strategies --- adjustment coefficient --- logarithmic asymptotics --- quadratic programming problem --- ruin probability --- two-dimensional Brownian motion --- spectrally negative Lévy process --- general tax structure --- first crossing time --- joint Laplace transform --- potential measure --- Laplace transform --- first hitting time --- diffusion-type process --- running maximum and minimum processes --- boundary-value problem --- normal reflection --- Sparre Andersen model --- heavy tails --- completely monotone distributions --- error bounds --- hyperexponential distribution --- reflected Brownian motion --- linear diffusions --- drawdown --- Segerdahl process --- affine coefficients --- spectrally negative Markov process --- hypergeometric functions --- capital injections --- bankruptcy --- reflection and absorption --- Pollaczek–Khinchine formula --- scale function --- Padé approximations --- Laguerre series --- Tricomi–Weeks Laplace inversion --- Lévy processes --- non-random overshoots --- skip-free random walks --- fluctuation theory --- scale functions --- capital surplus process --- dividend payment --- optimal control --- capital injection constraint --- spectrally negative Lévy processes --- reflected Lévy processes --- first passage --- drawdown process --- spectrally negative process --- dividends --- de Finetti valuation objective --- variational problem --- stochastic control --- optimal dividends --- Parisian ruin --- log-convexity --- barrier strategies --- adjustment coefficient --- logarithmic asymptotics --- quadratic programming problem --- ruin probability --- two-dimensional Brownian motion --- spectrally negative Lévy process --- general tax structure --- first crossing time --- joint Laplace transform --- potential measure --- Laplace transform --- first hitting time --- diffusion-type process --- running maximum and minimum processes --- boundary-value problem --- normal reflection --- Sparre Andersen model --- heavy tails --- completely monotone distributions --- error bounds --- hyperexponential distribution --- reflected Brownian motion --- linear diffusions --- drawdown --- Segerdahl process --- affine coefficients --- spectrally negative Markov process --- hypergeometric functions --- capital injections --- bankruptcy --- reflection and absorption --- Pollaczek–Khinchine formula --- scale function --- Padé approximations --- Laguerre series --- Tricomi–Weeks Laplace inversion